Download View File

Control Engineering
Lecture #3
19th March,2008
Models of Physical Systems




Two types of methods used in system modeling:
(i) Experimental method
(ii) Mathematical method
Design of engineering systems by trying and error
versus design by using mathematical models.
 Mathematical model gives the mathematical
relationships relating the output of a system to its
input.
Models of Electrical Circuits
 Resistance circuit: v(t) = i(t) R
 Inductance circuit:

Models of Electrical Circuits
 Capacitance circuit:
Models of Electrical Circuits
 Kirchhoff’ s voltage law:
The algebraic sum of voltages around any
closed loop in an electrical circuit is zero.
 Kirchhoff’ s current law:
The algebraic sum of currents into any
junction in an electrical circuit is zero.
Models of Electrical Circuits
 Example:
Transfer Function
 Suppose we have a constant-coefficient
linear differential equation with input f(t) and
output x(t).
 After Laplace transform we have
X(s)=G(s)F(s)
 We call G(s) the transfer function.
An Example
 Linear differential equation
 The Laplace transform is:
An Example
 Differential equation:
Characteristic Equation
Block Diagram and Signal Flow
Graphs
 Block diagram:
 Signal flow graph is used to denote graphically the transfer
function relationship:
 System interconnections
 Series interconnection
Y(s)=H(s)U(s) where H(s)=H1(s)H2(s).
 Parallel interconnection
Y(s)=H(s)U(s) where H(s)=H1(s)+H2(s).
 Feedback interconnection
An Example
 Parallel interconnection:
Another example:
Mason’s Gain Formula
 Motivation:
How to obtain the equivalent Transfer Function?
Ans: Mason’s formula
Mason’s Gain Formula
 This gives a procedure that allows us to find the
transfer function, by inspection of either a block
diagram or a signal flow graph.
 Source Node: signals flow away from the node.
 Sink node: signals flow only toward the node.
 Path: continuous connection of branches from
one node to another with all arrows in the same
direction.
 Forward path: is a path that connects a source to
a sink in which no node is encountered more than
once.
 Loop: a closed path in which no node is
encountered more than once. Source node
cannot be part of a loop.
 Path gain: product of the transfer functions of
all branches that form the path.
 Loop gain: products of the transfer functions
of all branches that form the loop.
 Nontouching: two loops are non-touching if
these loops have no nodes in common.
An Example
 Loop 1 (-G2H1) and loop 2 (-G4H2) are not
touching.
 Two forward paths:
More Examples:
P1  G1G2G3G4
L1  G2G3 H 2
L2  G3G4 H 1
L3  G1G2G3G4 H 3
  1  G2G3 H 2  G3G4 H 1  G1G2G3G4 H 3
M 1  G1G2G3G4
1  1
G1G2G3G4
G(s) 

Another Example: